# Mathematical Approach to Estimating the Main Epidemiological Parameters of African Swine Fever in Wild Boar

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## Abstract

**:**

_{0}) and effective (R

_{e}) reproduction numbers demonstrate that the ASF epidemic is declining and under control with an R

_{0}of 1.139 (95% confidence interval (CI) = 1.123–1.153) and R

_{e}of 0.802 (95% CI = 0.612–0.992). In the last phases of an epidemic, these estimates are crucial tools for identifying the intensity of interventions required to definitively eradicate the disease. This approach is useful to understand if and when the detection of residual seropositive WB is no longer associated with any further ASFV circulation.

## 1. Introduction

^{2}versus average of 5.2, SD of 0.62), the prevalent type of land as Mediterranean scrub, and with a very high level of compliance with the rules of the last ASF Eradication Plan (ASF-EP15-18), over 85% (mean = 85.8, 95% confidence interval (CI) of the mean = 73.4–98.2). For this purpose, the epidemiological situation of the north of the island during the last 10 years was studied in depth and an optimization analysis based on a stochastic model was conducted to validate the estimated epidemiological parameters.

## 2. Materials and Methods

#### 2.1. Sardinian Epidemiological Context

#### 2.2. Wild Boar Management Unit of Anglona-Gallura

^{2}, bordered by the sea; its territory is mainly characterized by hilly small plateaus of a volcanic or limestone nature. The estimates of the wild boar population average in Anglona-Gallura is approximately 13,550 animals, based on the wildlife management plan and as explained in previous studies [21,30,34]. The presence of free-ranging pigs, with notoriously high interaction rate with wild boar population in Sardinia [11,30,31], has never been detected within this area.

#### 2.3. Data Collection

^{®}, Ingenasa, Madrid, Spain), then positivity was confirmed through immunoblotting (IB) [15]. Serum samples were considered positive when they scored positive in both the screening (ELISA) and the confirmatory (IB) tests. Presence of ASF viral genome was assessed by real-time PCR [15,35].

#### 2.4. Estimates of the ASFV Force of Infection

#### 2.4.1. Method 1: Estimating λ by Contact Rate

#### 2.4.2. Method 2: Estimating λ Using a Catalytic Model

#### 2.4.3. Method 3: Estimating λ Based on Proportion of Infected

_{a}is the proportion of the population in the age group a, S

_{t}

_{,a}is the number of susceptible (ASF-tested negative) among those tested in age group a at time (t), and N

_{t}

_{,a}is the number of animals of each age group a that were ASF-tested [26]. The proportions a of wild boar population in each age group were defined in the literature as 0.63, 0.19, and 0.18 for young, subadult, and adult, respectively, considering intermediate environmental conditions [39,40]. This calculation was applied using data from the 2011–2012 and 2012–2013 hunting seasons. Subsequently, to generate a time-unit measure of one day for λ, the result was divided for about 30 days (1 month).

#### 2.5. Basic Reproduction Number (R_{0}) Estimation

_{0}) is the average of secondary infectious individuals that would result when one infectious individual is introduced for the first time into a completely susceptible population [41]. This parameter is able to quantify the spread of an infectious disease, predicting its speed, scale, and the level of herd immunity required to contain the disease. The higher the value of R

_{0}, the faster the epidemic progresses [36]. Assuming a well-mixed population, when a proportion p of the population is effectively protected from infection at time t (for t > 0), this parameter is known as the effective reproduction number (R

_{e}), which is related to R

_{0}by R

_{e}= (1–p) × R

_{0}[24]. When R

_{e}≤ 1, the epidemic is in decline and may be regarded as being under control at time t (and vice versa when R

_{e}> 1).

_{0}have been proposed, in this work, four different methods were applied based on the main assumption that individuals mix randomly in an endemic context.

#### 2.5.1. Method 1: Estimating R_{0} from the Doubling Time

_{0}(new infections per generation) and the infectious period (T) as:

#### 2.5.2. Method 2: Estimating R_{0} from the Force of Infection (λ)

_{0}parameter by can be computed based on the force of infection [26,42,43,44]:

#### 2.5.3. Method 3: Estimating R_{0} from the Proportion of Infected

_{0}) and end (S

_{f}) of the epidemic period (2011–2012 and 2012–2013 hunting seasons), N is the total population of ASF-tested animals and C is the number of cases (i.e., individuals who have been infected, seropositives), the R

_{0}parameterization proposed by Becker [45] was applied:

_{0}is defined by Equation (4) and S

_{f}is defined as S

_{0}minus the number of cases that occurred in 2011–2013 (= 4). The standard errors and 95% confidence intervals (95% CI) of the certainty of the last estimated R

_{0}were calculated as follow:

#### 2.5.4. Method 4: Estimating R_{0} from a Simple Susceptible-Exposed-Infectious-Recovered (SEIR) Model (Optimization Analysis)

_{0}estimations [28]. Assuming that the population is completely susceptible at the beginning of the epidemic, the model is described by the system of differential equations based on time (t):

_{0}to be validated, β was input to the final model as:

_{0}estimates were constructed by generating sets of realizations of the best-fit curve using parametric bootstrap [50]. The best estimation was evaluated based on the log-likelihood deviance [51].

_{0}value for representing the observed data (total number of infected and susceptible, peak time, and last virus reporting), a Monte Carlo simulation with 50,000 iterations after a burn-in for convergence of 10,000 observations was performed to find the optimal parameter. A scatter plot comparison was applied, according to Saltelli et al. [52].

#### 2.6. Effective Reproduction Number (R_{e}) Estimation

_{e}defines the number of subsequent cases at a specific time, to evaluate the current situation in Anglona-Gallura, the serological data from the last three hunting seasons (2017–2020) were used to calculate R

_{e}based on the typical formulation proposed by Diekmann et al. [53] and confirmed by several authors. Defining s as the proportion of the population that is susceptible at time t of the epidemic, the estimate of R

_{e}can be computed as:

_{e}were computed based on the different R

_{0}parameterizations, defining s

_{t}as 1–z

_{f}.

_{e}was optimized using a Bayesian inference of the stochastic SEIR model. This formulation considers that the probabilistic nature of the contagion assumes a negative binomial distribution. Thus, starting from the real-data derivate by periodic reports at time t, the change in the cumulative number of cases can be defined as: $\u2206C\left(t\right)=C\left(t\right)-C\left(t-\tau \right)$, where τ denotes the time interval between reports, which is equal to one year in this dataset. Consequently, via Bayes’ theorem, the distribution of the number of future cases can be predicted by applying the probabilistic formulation of a model based on the value of R

_{e}and ∆C(t) as:

_{e}] is the prior distribution of R

_{e}, the denominator is a normalization factor, and the notification of new cases in the next reports allows the estimation of the probability distribution function of R

_{e}as the posterior [54,55,56]. The model was iterated several times using the posterior distribution at time t (i.e., the probability R

_{e}distribution from previous reports) as the prior for new cases at t + τ. Uncertainty bounds (Bayesian credible intervals, BCI) were obtained by extracting the average and maximum-likelihood values of R

_{e}by its estimated distribution.

_{0}parametrizations were stochastically implemented in the software package Berkeley Madonna (Version 8.3.18, Regents, University of California, CA, USA). The Bayesian stochastic SEIR model was implemented using R software ‘BRugs’ (Version 3.6.2, R-Foundation for Statistical Computing, Vienna, Austria) and Open BUGS (version 3.2.3).

## 3. Results

_{0}to be 1.124 (95% CI = 1.103–1.145) during the first two hunting seasons. As reported in Table 3, the number of secondary cases R

_{0}calculated based on λ was described by a value of 1.165 (95% CI = 1.027–1.187). Based on the proportion of infected, an R

_{0}of 1.170 (95% CI = 1.009–1.332) was estimated for the first two years under study. The estimates of the reproduction number obtained by the three methods were found to be consistent with each other (in the range R ≈ 1–1.3, with overlapping CIs). The estimation method applied to the R

_{e}calculation for the last three years (2017–2020) showed a value of 0.802 (95% CI = 0.612–0.983).

_{0}ASF-specific for the WB-MU of Anglona-Gallura, four different curves describing the total number of infected animals were estimated by the optimization method with the SEIR models (Figure 6). The first model fitted using a value of R

_{0}of 1.124 described an infectious curve with an expected peak of 53 virus-infected wild boar at time point 729 (20 January 2014) and estimated the last peak at time 1315 (29 August 2015) (green line). The second SEIR fitted by the application of the λ value in the R

_{0}calculation (R

_{0}= 1.165) estimated a peak of 88 virus-positives at time 619 (2 October 2013) and the last at time point 1112 (7 February 2015) (blue line). The third system of differential equations implemented with an R

_{0}of 1.170, obtained by the number of susceptible animals, predicted a total of 93 infected animals at time 610 (23 September 2013) and the last virus-positive individual at time 1091 (17 January 2014; blue line).

_{0}obtained using the SEIR optimized method was ≈ 1.139 (95% CI = 1.123–1.153), with a peak at time point 687 (9 December 2013) with 64 infected animals, and the last positive animal detected at time point 1236 (11 June 2015) (black solid and dashed lines). Considering the scatter plot and log-likelihood deviation, the values of R

_{0}estimated by the third method (based on susceptible animals) seems to more accurately describe the observed data with an overall deviance of 132 (10 degrees of freedom). The Bayesian SEIR model estimated the overall average of the effective reproduction number as 0.923 (BCI = 0.812–1.033) with level of uncertainty between observed and estimated (maximum likelihood) measured by the width of the BCI, which decreased as case observations increased (Figure 7). At later times, R

_{e}< 1 (effective reproduction number = 0.769, BCI = 0.660–0.879) as a result of averaging periods in which the epidemic declined.

## 4. Discussion

_{0}at first virus detection (PCR-positive) in wild boar ranged from 1.24 to 1.70, whereas the R

_{e}estimation of 0.802 suggests that the disease is now under control.

_{0}values proposed in the literature for ASF vary widely and should be considered as context specific.

## 5. Conclusions

_{0}and R

_{e}revealed that at the end of the PCR-positive detection period, R

_{e}progressed to and remained below zero and the infection faded out spontaneously despite the presence of seropositive wild boars. However, even if the mathematical values of the epidemiological parameters indicate that the virus is fading out, the conclusions have to be further confirmed:

- (1)
- (2)
- The efficiency of passive surveillance proves that the absence of the virus (i.e., number of negative carcasses found with respect to the local wild boar population) is far from being known and standardized;
- (3)
- The absence of virus-positive animals among the hunted wild boar for a long time, as associated with serological data, provides valuable information for confirming that the disease is moving toward self-extinction; and
- (4)
- The seropositive and virus-negative animals that are still considered carriers despite their presence do not lead to any further virus detection, as confirmed by this study and in other similar epidemiological situation (e.g., in part of Estonia and Latvia). Notably, ASFV-specific antibodies in a wild boar population were found several years after the fading out of the virus [20,60,61,62,63].

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**A**) Sardinian African swine fever (ASF) trend in domestic pigs is described as estimation (blue line) and 95% confidence interval (CI; gray lines), for the years 2011–2020. The secondary axis measures the percentage of compliance in domestic pig farms (with the rules of ASF-EP15-18), represented as box plots. The whiskers represent the minimum and maximum values and the ends of the rectangle represent the first and third quartiles, whereas the segment inside the rectangle is the median value and the dots represent the outlier values. The last box plot related to 2020 refers to the first six months of the year. (

**B**) Sardinian ASF disease prevalence trend in wild boar. The virus-positive trend is represented by the green line and 95% CI (gray lines). The seropositive trend is represented by the blue line and 95% CI (gray lines). Data are expressed as percentage.

**Figure 2.**(

**a**) Overall seasonality of African swine fever events in the Anglona area of the Sardinian region (

**b**) in domestic pig farms and (

**c**) in the wild boar population, from January 2012 to December 2019 corrected for number of observation months.

**Figure 3.**Contextualization of the Sardinian data based on wild boar management unit. The wild boar infection zone (IZ) is indicated by the red line.

**Figure 4.**(

**a**–

**h**) ASF virus prevalence trend in wild boar based on 2011–2020 data of the four wild boar management units included in the infected zone. The trends are represented as African swine fever virus and seroprevalence trends for the wild boar management units of (

**a**,

**b**) Anglona-Gallura, (

**c**,

**d**) Goceano-Gallura, (

**e**,

**f**) Nuoro-Baronia, and (

**g**,

**h**) Gennargentu-Ogliastra. Data are expressed as percentage of prevalence.

**Figure 5.**Number of ASF virus (ASFV) PCR-positive and seropositive animals detected in Anglona-Gallura during 2011–2020, divided into young, subadult, and adult categories.

**Figure 6.**Number of ASFV infectious animals estimated in Anglona-Gallura by the four SEIR models fitted for 1500 time points. The blue line represents the curve of infectious individuals estimated by the R

_{0}(a) based on doubling time (red line), R

_{0}(b) based on force of infection (green line), R

_{0}(c) based on the proportion of infected (blue line), and R

_{0}(d) based on SEIR model (optimization). The dashed lines represent the 95% CI of the optimized R

_{0}.

**Figure 7.**Sequential Bayesian estimation of the full distribution of effective reproduction number (R

_{e}) leads to the estimation of its maximum-likelihood value (blue dots) and BCIs (black lines). Upper limit of the estimate is below 1, indicating virus fade out.

**Table 1.**Summary of baseline database created for the active surveillance in Anglona-Gallura area, including time of hunting season (from 2011 to 2020), wild boar age classes (young, subadult, and adult), the number of wild boar tested for ASF virus presence or antibody presence, and the virus prevalence or seroprevalence. Data are presented as number and percentage, or number and prevalence with 95% confidence intervals.

Hunting Season | Wild Boar Age Classes | Wild Boar Virologically ^{1} Tested | Virus Prevalence | Wild Boar Serologically ^{2} Tested | Seroprevalence ^{3} |
---|---|---|---|---|---|

n (%) | (95% CI) | n (%) | (95% CI) | ||

2011–2012 | Young | 40 (33.6) | 0 (0–0) | 272 (28.1) | 0 (0–0) |

Subadult | 47 (39.5) | 0 (0–0) | 277 (28.6) | 0 (0–0) | |

Adult | 32 (26.9) | 6.4 (0.1–16.2) | 418 (43.3) | 0 (0–0) | |

Total | 119 | 0.8 (0.2–4.6) | 967 | 0 (0–0) | |

2012–2013 | Young | 21 (17.2) | 0 (0–0) | 131 (19.5) | 0.8 (0.0–4.2) |

Subadult | 34 (27.9) | 0 (0–0) | 171 (25.5) | 0.6 (0.0–3.2) | |

Adult | 67 (54.9) | 1.5 (0.0–8.0) | 369 (55.0) | 0.5 (0.1–1.9) | |

Total | 122 | 0.8 (0.0–4.5) | 671 | 0.6 (0.2–1.5) | |

2013–2014 | Young | 33 (6.8) | 0 (0–0) | 55 (4.5) | 5.5 (1.1–15.1) |

Subadult | 191 (39.2) | 3.1 (1.2–6.7) | 368 (30.3) | 1.4 (0.4–3.1) | |

Adult | 263 (54.0) | 4.2 (2.1–7.3) | 792 (65.2) | 0.8 (0.3–1.6) | |

Total | 487 | 3.5 (2.0–5.5) | 1215 | 1.1 (0.6–1.9) | |

2014–2015 | Young | 33 (10.1) | 0 (0–0) | 80 (5.3) | 0 (0–0) |

Subadult | 98 (29.9) | 0 (0–0) | 410 (27.3) | 0.5 (0.1–1.7) | |

Adult | 197 (60.0) | 1.5 (0.3–4.4) | 1014 (67.4) | 3.5 (2.0–4.2) | |

Total | 328 | 0.9 (0.2–2.6) | 1504 | 2.1 (1.4–3.0) | |

2015–2016 | Young | 53 (7.8) | 0 (0–0) | 95 (5.8) | 1.1 (0.0–5.7) |

Subadult | 182 (27.0) | 0 (0–0) | 420 (25.4) | 0.2 (0.0–1.3) | |

Adult | 440 (65.2) | 0.7 (0.1–2.0) | 1137 (68.8) | 2.9 (2.0–4.0) | |

Total | 675 | 0.4 (0.0–1.3) | 1652 | 2.1 (1.5–2.9) | |

2016–2017 | Young | 64 (7.1) | 0 (0–0) | 117 (4.9) | 0 (0–0) |

Subadult | 244 (27.1) | 0 (0–0) | 657 (27.7) | 0.9 (0.3–2.0) | |

Adult | 591 (65.8) | 0 (0–0) | 1600 (67.4) | 1.4 )0.9–2.1) | |

Total | 899 | 0 (0, 0–0) | 2374 | 1.2 (0.8–1.7) | |

2017–2018 | Young | 71 (6.1) | 0 (0–0) | 132 (5.9) | 0 (0–0) |

Subadult | 364 (31.3) | 0 (0–0) | 679 (30.6) | 0.6 (0.2–1.5) | |

Adult | 729 (62.6) | 0 (0–0) | 1408 (63.5) | 0.6 (0.2–1.1) | |

Total | 1164 | 0 (0, 0–0) | 2219 | 0.5 (0.3–0.9) | |

2018–2019 | Young | 78 (6.1) | 0 (0–0) | 97 (4.2) | 0 (0–0) |

Subadult | 290 (22.8) | 0 (0–0) | 531 (22.7) | 0 (0–0) | |

Adult | 901 (71.1) | 0 (0–0) | 1709 (73.1) | 0.5 (0.2–0.9) | |

Total | 1269 | 0 (0, 0–0) | 2337 | 0.3 (0.1–0.7) | |

2019–2020 | Young | 53 (4.0) | 0 (0–0) | 92 (3.9) | 0 (0–0) |

Subadult | 339 (25.8) | 0 (0–0) | 599 (25.3) | 0.2 (0.0–0.9) | |

Adult | 921 (70.2) | 0 (0–0) | 1674 (70.8) | 0.2 (0.1–0.6) | |

Total | 1313 | 0 (0, 0–0) | 2365 | 0.2 (0.1–0.5) |

^{1}Virus presence was assessed by real-time PCR, quantitative PCR, or Malmquist test.

^{2}Wild boar serologically tested were those tested with at least the screening ELISA test (Ingezim PPA Compac

^{®}, Ingenasa, Madrid, Spain), and eventually confirmed (if positive) with immunoblotting (IB), in accordance with the Manual of Diagnostic Tests and Vaccines for Terrestrial Animals [15].

^{3}Serum samples were considered positive when they scored positive in both the screening (ELISA) and the confirmatory (IB) tests.

**Table 2.**Estimate of standard error of the mean (SEM), 95% CIs of the mean, and width of CI of the original data observed during the Sardinian hunting season.

Variable | N | Mean | SEM | 95% CI of Mean | Width of the CI |
---|---|---|---|---|---|

Week 1a | 23 | 39.874 | 0.859 | 38.189–41.558 | 3.368 |

Week 1b | 21 | 8.056 | 0.350 | 7.369–8.742 | 1.373 |

Week 2a | 22 | 42.073 | 0.943 | 40.223–43.922 | 3.699 |

Week 2b | 24 | 10.632 | 0.448 | 9.753–11.511 | 1.758 |

Week 3a | 21 | 40.000 | 0.878 | 38.278–41.722 | 3.444 |

Week 3b | 22 | 9.476 | 0.391 | 8.708–10.243 | 1.535 |

Week 4a | 24 | 38.000 | 0.973 | 36.092–39.908 | 3.816 |

Week 4b | 22 | 11.000 | 0.471 | 10.076–11.924 | 1.848 |

**Table 3.**Summary of the parameters estimated using the different methods, expressed as estimates and 95% confidence intervals (95% CIs).

Parameter | Definition | Method | Estimate | 95% CI |
---|---|---|---|---|

λ | Force of infection (daily) | Based on contact rate β | 0.0056 | 0.0016–0.0092 |

Catalytic model | 0.0024 | 0.0013–0.0035 | ||

Based on average age at infection A | 0.0012 | 0.0004–0.0021 | ||

A | Average age at infection (monthly) | Life expectancy | 28.5 | 28.1–29.9 |

s | Proportion of susceptible/receptive population | Serology by age | 0.952 | 0.937–0.991 |

R_{0} | Basic reproduction number | Doubling time | 1.124 | 1.103–1.145 |

Based on force of infection λ | 1.165 | 1.027–1.301 | ||

Proportion of infected | 1.170 | 1.009–1.332 | ||

SEIR model (optimization) | 1.139 | 1.123–1.153 | ||

R_{e} | Effective reproduction number | Average value based on basic reproduction number R_{0} | 0.802 | 0.612–0.992 |

Bayesian SEIR model (optimization) (BCI) | 0.923 | 0.812–1.033 | ||

f | Infection rate | 1/average pre-infectious period [46,47,48] | 0.28 | – |

r | Recovery rate | 1/infectious period [46,47,48] | 0.15 | – |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Loi, F.; Cappai, S.; Laddomada, A.; Feliziani, F.; Oggiano, A.; Franzoni, G.; Rolesu, S.; Guberti, V. Mathematical Approach to Estimating the Main Epidemiological Parameters of African Swine Fever in Wild Boar. *Vaccines* **2020**, *8*, 521.
https://doi.org/10.3390/vaccines8030521

**AMA Style**

Loi F, Cappai S, Laddomada A, Feliziani F, Oggiano A, Franzoni G, Rolesu S, Guberti V. Mathematical Approach to Estimating the Main Epidemiological Parameters of African Swine Fever in Wild Boar. *Vaccines*. 2020; 8(3):521.
https://doi.org/10.3390/vaccines8030521

**Chicago/Turabian Style**

Loi, Federica, Stefano Cappai, Alberto Laddomada, Francesco Feliziani, Annalisa Oggiano, Giulia Franzoni, Sandro Rolesu, and Vittorio Guberti. 2020. "Mathematical Approach to Estimating the Main Epidemiological Parameters of African Swine Fever in Wild Boar" *Vaccines* 8, no. 3: 521.
https://doi.org/10.3390/vaccines8030521